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Blowup properties for a degenerate parabolic system coupled via nonlinear boundary flux
Boundary Value Problems volume 2015, Article number: 188 (2015)
Abstract
In this paper, we study the simultaneous and nonsimultaneous blowup problem for a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain hypotheses on the initial data and parameters, we prove that nonsimultaneous blowup is possible. Moreover, we get some conditions on which simultaneous blowup must occur, as well as the nonsimultaneous blowup conditions for every initial data. Furthermore, we get a result on the coexistence of both simultaneous and nonsimultaneous blowups.
Introduction and main results
In this paper, we study the nonlinear parabolic system
with nonlinear coupling boundary conditions
and the initial data
where \(m,n>1\), \(\alpha, \beta, p_{1}, q_{2}\geq0\), \(p_{2}, q_{1}>0\), the initial conditions \(u_{0},v_{0}\geq\delta>0\), are continuous bounded and satisfy
Remark 1.1
Combining (1.4) with the comparison principle, we can actually get following relationships:
The reactiondiffusion system (1.1)(1.3) can be used to describe heat transfer in a mixed medium with absorptions and nonlinear boundary flux, and some chemical reaction processes with the slow diffusion phenomenon (see [1, 2]).
In [3], Song and Zheng considered the blowup conditions of the following problem:
where Ω is a bounded domain in \(R^{N}\), \(m, n >0\), \(\alpha_{i}, \beta_{i}, p_{i}, q_{i}\geq0\). If we let \(u^{m}=\overline{u}\), \(v^{n}=\overline{v}\), \(N=1\), \(p_{1}=q_{1}=0\), we just get system (1.1)(1.3). From the conclusions in [3], we know that the solution of system (1.1)(1.3) blows up if and only if one of the following conditions holds:

(1)
\(\max\{\alpha, p_{1}\}>1\),

(2)
\(\max\{\beta, q_{2}\}>1\),

(3)
\(p_{2}q_{1}>(1p_{1})(1q_{2})\).
In this case we can only have
However, from the above result, we cannot show that blowup is simultaneous or nonsimultaneous. The blowup rate is not known yet.
Recently, the simultaneous and nonsimultaneous blowup problems of parabolic systems have been widely considered by many authors [4–14]. For example, when \(m=n=1\), \(\lambda_{i}=0\), Pinasco and Rossi [6] considered the system of heat equations coupled via a nonlinear boundary flux with \(\Omega\subset R^{N}\) and found that u blows up at time T and v remains bounded up to time T for certain initial data if and only if \(p_{1}>1\) and \(p_{2}< p_{1}1\).
When \(\lambda_{i}=0\), \(\alpha=\beta=0\), \(m>0\), \(n>0\), Brändle et al. [4] studied problem (1.1)(1.3) and proved that the nonsimultaneous blowups occur if and only if
They also found some conditions under which u blows up and v remains bounded for every initial data.
When \(m=n=1\) and \(\lambda_{i}<0\), Zheng and Qiao [10] also got the sufficient and necessary conditions of nonsimultaneous blowup.
In this paper, by using a modification of methods in [4] and [10] we will focus on the simultaneous and nonsimultaneous blowup problems to (1.1)(1.3), and we get our main results as follows.
Theorem 1.1
When \(\lambda_{i}>0\), if \(\max\{\alpha, p_{1}\}>m\) and
then there exists initial data \((u_{0}, v_{0})\) such that u blows up at a finite time T, while v remains bounded up to T.
Theorem 1.2
When \(\lambda_{i}>0\) and \(\max\{\alpha, p_{1}\}>m\), if u blows up at time T and v remains bounded up to T, then (1.5) holds.
Theorem 1.3
When \(\lambda_{i}<0\), if \(p_{1}>m\) and \(2p_{1}m>\max\{2p_{2}+1, \alpha\}\), then there exists initial data \((u_{0}, v_{0})\) such that u blows up at a finite time T, while v remains bounded up to T.
Remark 1.2
When \(m=n=1\) and \(\lambda_{i}<0\), Theorem 1.3 is consistent with Theorem 1 in [10].
By interchanging the roles of u and v, we get the following results.
Corollary 1.4
When \(\lambda_{i}>0\),

(i)
if \(\max\{\beta,q_{2}\}>n\) and
$$ \max\{2q_{2}n,\beta\}>2q_{1}+1, $$(1.6)then there exists initial data \((u_{0}, v_{0})\) such that v blows up at a finite time T, while u remains bounded up to T.

(ii)
When \(\max\{\beta,q_{2}\}>n\), if v blows up at a finite time T and u remains bounded up to T, then (1.6) holds.
Corollary 1.5
When \(\lambda_{i}<0\), if \(q_{2}>n\) and \(2q_{2}n>\max\{2q_{1}+1, \beta\}\), then there exists initial data \((u_{0}, v_{0})\) such that v blows up at a finite time T, while u remains bounded up to T.
Theorem 1.6
When \(\lambda_{i}>0\), if \(\max\{\alpha, p_{1}\}>m\) and \(2p_{2}+1\geq\max\{2p_{1}m,\alpha\}\), then for any initial data \((u_{0}, v_{0})\) the solution \((u,v)\) to (1.1)(1.3) blows up simultaneously at a finite time T.
Theorem 1.7
When \(\lambda_{i}>0\), if \(\max\{\alpha, p_{1}\}>m\) and (1.5) holds, then the set of initial data such that u blows up and v remains bounded is open in the \(L^{\infty}\) topology. If \(\max\{\beta, q_{2}\}>n\) and (1.6) holds, then the set of initial data such that v blows up and u remains bounded is open in \(L^{\infty}\) topology.
Next, inspired by [9, 12], we consider the coexistence of both simultaneous and nonsimultaneous blowups, and we get the blowup rate if simultaneous blowups occur.
Theorem 1.8
When \(\lambda_{i}>0\), if
and
then there may occur both simultaneous and nonsimultaneous blowups. Moreover, if \((u, v)\) blow up simultaneously, then there exist positive constants c, C such that
where
Finally, let us show that, under certain conditions, blowup is always nonsimultaneous.
Theorem 1.9
When \(\lambda_{i}>0\), if
and
then u blows up and v remains bounded for any initial data.
Theorem 1.10
When \(\lambda_{i}>0\), if \(\max\{\alpha, p_{1}\}>m\), \(\max\{\beta, q_{2}\}\leq1\) and (1.5) holds, then u blows up and v remains bounded for any initial data.
The rest of this paper is organized as follows. In the next section, we first get some blowup results of scalar problems, and using them to prove Theorems 1.11.3. In Section 3, we consider the coexistence of both simultaneous and nonsimultaneous blowups, Theorems 1.7, 1.8 are proved. In Section 4, we show that nonsimultaneous blowups always occur, and we prove Theorems 1.9 and 1.10.
Nonsimultaneous and simultaneous blowup
In this section, in order to prove Theorems 1.11.3, we will start with the following problem:
where \(l>1\), \(q, p>0\), and \(\lambda_{1}\in R\), \(h(t)\) is a continuous, bounded, nondecreasing, and strictly positive function.
When \(h=1\), \(\lambda_{1}=1\), problem (2.1) has been already studied in [15–17]. There, it is shown that solutions blow up if and only if
By the comparison principle, when \(\lambda_{1}>0\), the solutions of (2.1) blow up if and only if the same restriction.
When \(\lambda_{1}<0\), we know from [18], if \(p>l\) and \(q<2pl\), that the solutions of (2.1) blow up for large initial data.
We give the blowup rate of the solution of (2.1) in the following two lemmas.
Lemma 2.1
Let u be a solution of (2.1) with \(\lambda_{1}>0\), if \(\max\{q,p\}>l\), then there exist positive constants c and C, such that

(i)
When \(2p>q+l\), \(\sigma=\frac{1}{2pl1}\).

(ii)
When \(2p< q+l\), \(\sigma=\frac{1}{q1}\).

(iii)
When \(2p=q+l\), \(\sigma=\frac{1}{q1}=\frac{1}{2pl1}\).
Proof
Notice that if case (i) holds, by \(\max\{q, p\}>l\) and \(2p>q+l\), we get \(p>l\). If case (ii) holds, it follows from \(\max\{q, p\}>l\) and \(2p< q+l\) that \(q>l\). Thus, by a similar proof to that of Theorem 3.1 in [15], we can prove easily Lemma 2.1. We omit it here. □
Lemma 2.2
When \(\lambda_{1}<0\), \(p>l\), and \(2p>q+l\). If u be a solution of (2.1), then there exist positive constants c and C, such that
where \(\sigma=\frac{1}{2pl1}\).
Proof
The proof is similar to that of Theorem 1.2 in [18]. We omit it here. □
Furthermore, we need to consider the following problem.
where \(k_{i}, s>0\) and \(\lambda_{2}\in R\).
Lemma 2.3
If \(s<1/2\), then given \(v_{0}\) there exists T small enough such that the solution of (2.4) verifies
Proof
For given \(v_{0}\), we denote
let \(t_{0}\) be the first time that
If such \(t_{0}\) does not exist, v will remain bounded with \(v<2K\), the conclusion follows.
Let \(\Gamma(x,t)\) be the fundamental solution of the heat equation in \([0,1]\), so
It is well known that Γ satisfies (see [19]),
By Green’s identity of (2.4), we have
where \(0\leq z< t< T\), \(0< x<1\). With \(z=0\), and \(x\rightarrow1\), it follows that
Furthermore, we have
(i) If \(\lambda_{2}<0\), (2.5) becomes
Since \(s<1/2\), for \(N=2^{k_{2}}K^{k_{2}1}+3\), if we choose T sufficiently small enough, \(\int^{t_{0}}_{0}(t_{0}\tau)^{s1/2}\,d\tau\) can be smaller than \(2\sqrt{\pi}/NC\), also \(\sqrt{t_{0}}\leq\sqrt{T}\leq1/(NC^{*})\). Thus
since \(v(1,t_{0})=2K\),
and hence
this is a contradiction.
(ii) If \(\lambda_{2}>0\), for
if we choose T sufficiently small enough, \(\int^{t_{0}}_{0}(t_{0}\tau)^{s1/2}\,d\tau\) can be smaller than \(2\sqrt{\pi}/NC\), also \(\sqrt{t_{0}}\leq\sqrt{T}\leq1/(NC^{*})\). Thus
and hence
this is also a contradiction. □
Lemma 2.4
For given \(v_{0}\), let \(v(x,t)\) be the solution of the following problem:
where \(n>1\), \(\beta, q_{2}, s>0\), and \(\lambda_{2}\in R\). If \(s<1/2\), then there exists T small enough such that \(v(x,t)\) satisfies
Proof
Given \(v_{0}\), let z be a solution of (2.4), with \(z_{0}(x)=v_{0}^{n}(x)\), \(k_{1}=\frac{\beta}{n}\), \(k_{2}=\frac{q_{2}}{n}\). A constant \(C(z_{0})=(n(2\z_{0}\_{\infty})^{\frac{n1}{n}})^{1}\) in front of \(z_{t}\). Let \(\overline{v}=z^{1/n}\), then
since \(s<1/2\), from Lemma 2.3, we know that a small enough T can be found, such that
z is also bounded away from zero and \(z_{t}>0\), so we have
Therefore v̅ is a supersolution of problem (2.7). Hence
□
Lemma 2.5
If \(n>1\) and \(s\geq1/2\), then every nonnegative, nontrivial solution of the following problem blows up at time T,
Proof
It is an immediate conclusion of Theorem 3.2(i) in [4]. □
Proof of Theorem 1.1
First, we consider \(\underline{u}\) to be a solution of the following problem:
Since \(\max\{\alpha,p_{1}\}>m>1\), \(\underline{u}\) blows up at finite time \(T'\). From Lemma 2.1, we also have the estimate
In particular, \(u_{0}(x)=u(x,0)\leq C(T')^{\sigma}\), that is, \(T'\leq(\frac{C}{u_{0}(x)})^{1/\sigma}\).
Since \(v\geq\delta>0\), for all \((x,t)\in[0,1]\times[0,T)\), u is a supersolution of \(\underline{u}\). Then u also blows up, and the blowup time T is smaller than \(T'\). Thus we have \(T\leq T'\leq(\frac{C}{u_{0}(x)})^{1/\sigma}\). As \(\sigma>0\), we can choose the initial data \(u_{0}(x)\) large enough such that T is small.
Given \(v_{0}\), we denote
We claim that there exists \(u_{0}\) large enough such that \(v<3K\) for all \((x,t)\in [0,1]\times[0,T]\). If this claim is not true, there exists \(0< t_{0}< T\), which is the first time \(v(1,t_{0})=3K\). So when \(t\in[0,t_{0}]\), \(v\leq3K\), when \(t\in[t_{0},T]\), \(v\geq3K\). We denote a cutoff function
obviously, ṽ is a bounded, continuous, and nondecreasing function.
Let ũ be a solution of following problem:
ũ blows up at T̃, by Lemma 2.1,
As \(\widetilde{v}\leq v\) for \((x,t)\in[0,1]\times[0,T]\), ũ is a subsolution of u, thus we have \(T\leq\widetilde{T}\). Therefore,
Now we consider the equations of v,
So v is a subsolution of (2.7). Let v̅ be the solution of (2.7), where \(s=p_{2}\sigma\).

(i)
If \(2p_{1}>\alpha+m\), from (1.5) we have \(2p_{1}m>2p_{2}+1\), thus \(s=p_{2}\sigma=\frac{p_{2}}{2p_{1}m1}<1/2\).

(ii)
If \(2p_{1}<\alpha+m\), from (1.5) we have \(\alpha>2p_{2}+1\), also get \(s=\frac{p_{2}}{\alpha1}<1/2\).

(iii)
If \(2p_{1}=\alpha+m\), from (1.5) we have \(2p_{1}m=\alpha=2p_{2}+1\), and we get
$$s=\frac{p_{2}}{\alpha1}=\frac{p_{2}}{2p_{1}m1}< 1/2. $$
By Lemma 2.4, we have \(\overline{v}(1,t_{0})<2K\). But on the other hand, we have
which is a contradiction; we conclude that v remains bounded up to time T. □
Proof of Theorem 1.2
Let \(h(t)=v(1,t)\). Since \(\max\{\alpha, p_{1}\}>m\), and v is a bounded, continuous nondecreasing function in \([0, T]\), by Lemma 2.1, we know
Since \(v\geq\delta\), v satisfies
So v is a supersolution of (2.8). Let \(\underline{v}\) be the solution of (2.8) with \(s=p_{2}\sigma\). By Lemma 2.5, we have \(s<1/2\), otherwise \(\underline{v}\) will blow up, and v cannot be bounded.

(i)
If \(2p_{1}>\alpha+m\), from \(s=\frac{p_{2}}{2p_{1}m1}<1/2\), we have \(2p_{1}m>\max\{2p_{2}+1, \alpha\}\).

(ii)
If \(2p_{1}<\alpha+m\), from \(s=\frac{p_{2}}{\alpha1}<1/2\), we have \(\alpha>\max\{2p_{2}+1, 2p_{1}m\}\).

(iii)
If \(2p_{1}=\alpha+m\), from \(s=\frac{p_{2}}{\alpha1}=\frac{p_{2}}{2p_{1}m1}<1/2\), we have \(2p_{1}m=\alpha>2p_{2}+1\).
From the previous three cases, we have (1.5). □
Proof of Theorem 1.3
By using the blowup rate estimate in Lemma 2.2 instead of the estimate in Lemma 2.1, Theorem 1.3 can be proved by the same steps as in the proof of statement (i) in Theorem 1.1. We omit it here. □
Proof of Theorem 1.6
Under our assumptions u must blow up. If there exists initial data that u blows up at finite time T, while v remain bounded up to that time, from Theorem 1.2 we have (1.5) holds, which is a contradiction. □
Coexistence of simultaneous and nonsimultaneous blowups
In this section, we consider the coexistence of both simultaneous and nonsimultaneous blowups, and we prove Theorems 1.7 and 1.8.
Proof of Theorem 1.7
Let \((u, v)\) be a solution of (1.1)(1.3) with initial data \((u_{0}, v_{0})\) such that u blows up at T while v remains bounded, that is, \(v\leq C\). We only need to find a \(L^{\infty}\)neighborhood of \((u_{0}, v_{0})\) such that any solution \((\hat{u}, \hat{v})\) of (1.1)(1.3) with initial data \((\hat{u_{0}},\hat{v_{0}})\) in this neighborhood maintains the property that u blows up while v remains bounded.
In fact, as the solutions of (1.1)(1.3) are bounded up to time \(T\varepsilon\), the continuity of solution respect to initial conditions holds up to \(T\varepsilon\), which means for any \(\epsilon_{0}\), there is a \(\delta_{0}\), \(\(\hat{u_{0}}, \hat{v_{0}})(u_{0}, v_{0})\_{\infty}<\delta_{0}\), such that \(\(\hat{u}, \hat{v})(u, v)\_{\infty}<\epsilon_{0}\). When we let \(\epsilon_{0}=1\), there is a \(\delta_{0}\), such that \(\(\hat{u}, \hat{v})(u, v)\_{\infty}<1\), so we get \(\\hat{u}u\_{\infty}<1\) and \(\\hat{v}v\_{\infty}<1\). Moreover, u becomes large at time \(T\varepsilon\) and \(v\leq C\) up to \(T\varepsilon\), so have found a neighborhood of \((u_{0},v_{0})\) in \(L^{\infty}\) such that, if \((\hat{u}, \hat{v})\) has initial data in such neighborhood, then û becomes large at time \(T\varepsilon\) and \(\hat{v}\leq C+1\) up to \(T\varepsilon\). The argument in the proof of Theorem 1.1 allows us to conclude that û blows up and v̂ remains bounded, if we consider time \(T\varepsilon\) as the initial time. □
Proof of Theorem 1.8
Under our assumptions, from Theorem 1.1, we know that the set of \((u_{0}, v_{0})\) such that u blows up and v remains bounded is nonempty. From Corollary 1.4(i), we also know the set of initial data for v blowing up and u being bounded is nonempty. Moreover, Theorem 1.7 concludes that such sets are open. Clearly, the two open sets are disjoint. That is to say, there exist \((u_{0}, v_{0})\) such that u and v blow up simultaneously at a finite time T.
Next, we will get a simultaneously blowup rate, if there are simultaneous blowups. Define
Set
where a, b, c, d are positive functions of t defined as follows:
Then we have
Moreover, \((\varphi_{M}, \psi_{N})\) satisfies the following problem:
where
Obviously, \(a, b, c, d, k_{M}, k_{N}\rightarrow0\) as \(t\rightarrow T\).
We claim that there exist positive constants c and C, such that
for every small a.
Here, we only consider the estimate of \((\varphi_{M})_{s}(0,0)\); that of \((\psi_{N})_{s}(0, 0)\) can be proven in a similar way. Given \(\{\varphi_{M_{j}}\}\), there is a continuous function φ and a subsequence, which we denote again by \(\{\varphi_{M_{j}}\}\), such that \(\varphi_{M_{j}}\rightarrow\varphi\) as \(M\rightarrow\infty\). Therefore, there exists a neighborhood of \((0, 0)\), U, such that \(\varphi>1/2\) in U. Since we have uniform convergence in U̅ (we can assume that U̅ is compact), for j large enough, we have \(1/4\leq\varphi_{M_{j}}\leq1\) in U̅. Thus the functions \(\varphi_{M_{j}}\) are solutions of uniformly parabolic equations in U̅, using the Schauder estimates
the upper bound in (3.1) follows immediately. To obtain the lower estimate, assume for finding a contradiction that there exists a sequence \(\{\varphi_{M_{j}}, \psi_{N_{j}}\}\) such that \((\varphi_{M_{j}})_{s}(0,0)\rightarrow0\). Since \(\varphi_{M_{j}}\rightarrow\varphi\), \(\psi_{N_{j}}\rightarrow\psi\), \(0<\varphi, \psi\leq1\), \(\varphi(0,0)=\psi(0,0)=1\), \(\frac{\partial\varphi}{\partial s},\frac{\partial\psi}{\partial s}\geq0\), and φ satisfies
\(w=\varphi_{s}\) satisfies
On the other hand, \((\varphi_{M_{j}})_{s}(0,0)\rightarrow\varphi_{s}(0,0)\), so \(w(0,0)=\varphi_{s}(0,0)=0\), since \(u_{t}\geq0\), \(w=\varphi_{s}\geq0\), hence w has a minimum at \((0,0)\), and by Hopf’s lemma, \(w\equiv0\), which implies φ does not depend on s. Thus \(\varphi=\varphi(y)\) satisfies
so \((\varphi^{m})(y)\geq1+y\), which contradicts the fact that \(0\leq\varphi\leq1\); thus (3.1) holds.
Rewriting the estimate in terms of M and N, we have
and hence
Integrating the above inequality on \((t, T)\) we obtain
Integrating on \((t, T)\), the estimate of v follows. Through a similar computation the estimate of u can be given. □
Nonsimultaneous blowups always happen
In this section, we will show that under certain conditions nonsimultaneous blowups always occur, and we will prove Theorems 1.9 and 1.10.
Proof of Theorem 1.9
Assume that there is an initial data \((u_{0}, v_{0})\), such that u and v both blow up at time T. Following the steps of the proof of Theorem 1.8, we get (3.2). Thus
First, we assume that \(2q_{1}2q_{2}+n+1>0\), after a straightforward computation we obtain
As \(2p_{2}2p_{1}+m+1<0\), and \(2q_{1}2q_{2}+n+1>0\), we obtain a contradiction with the assumption of simultaneous blowups.
If \(2q_{1}2q_{2}+n+1=0\), we have
also a contradiction. □
Proof of Theorem 1.10
It is well known that under the assumptions, v cannot blow up without the help of u. So the blowup time of v cannot be larger than that of u. u blows up at finite time T, by considering the solution at time \(T\varepsilon\) as initial data, we may assume that the blowup time is as small as desired. Following the same steps of the proof of Theorem 1.1, the conclusion follows. □
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Xu, S. Blowup properties for a degenerate parabolic system coupled via nonlinear boundary flux. Bound Value Probl 2015, 188 (2015). https://doi.org/10.1186/s1366101504497
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DOI: https://doi.org/10.1186/s1366101504497
MSC
 5B33
 35K65
 35K55
Keywords
 nonsimultaneous blowup
 blowup rate
 nonlinear diffusion
 nonlinear boundary flux